What is the best isotropic approximation of an anisotropic stiffness tensor? How can we quantify isotropy with a single numerical index? This article gives a precise meaning to these questions by introducing an orthogonal projector from the stiffness tensors space onto the isotropic, tensors subspace. Moreover, explicit expressions are derived for the Lamé parameters of this approximation, when the data are either in coordinate-free form or in Voigt’ s notation. Thus it is possible to unravel the effect of anisotropy on time-harmonic plane wave propagation. Detailed computations are shown in the case of ANNIE shales, which are transversely isotropic, by way of example. Finally, the isotropic approximations of 44 shales are computed, with the result that the anisotropy index is less than ten percent for just five of them. Potential applications of this approach to rock physics, seismic exploration, reservoir engineering, and seismology are briefly outlined.